Editing DarkMatter/CK2015 (section)

==A Potential Problem==
After sleeping on this last night, I realized that there is a potential problem with this proposed explanation of flat rotation curves in galaxy disks.  While C&amp;K plan to eventually use an SCF code to generate full models of their new equilibrium disks with "flat" rotation curves, the current paper does not yet discuss the vertical structure of these disks.  I am confident that they will find that their disks are pretty thick, vertically, and my recollection is that (HI) observations of edge-on spirals show that, in reality, the disks are vertically quite thin.  The basic realization is that, anywhere the rotation curve deviates significantly from Keplerian, that means that (radially at least) there is considerable support from pressure gradients &#8212; as GK15 explain.  But, since the pressure in ''gaseous'' configurations is isotropic, this means that in these regions of the disk, the ''vertical'' disk structure must also display a significant pressure support and, hence, be pretty thick.  Maybe the (HI) disks ''are'' vertically thick in terms of volume density but, when integrated along the line of sight, the surface density projections make the disks look thin.  At any rate, I think that the paper needs to address this question.

But wait!  After a bit more thought, I'm remembering that the rotationally flattened isothermal configurations (with flat rotation curves) generated by Hayashi, Narita &amp; Miyama (1982) can be thin as well as fat; it all depends on the ratio of the sound speed to the rotation velocity.  So maybe there isn't a vertical thickness problem.  Nevertheless this issue may need to be directly addressed in the paper.

Just for fun, I decided to use the VisTrails rendering tool to display images of some of the analytically prescribed HNM82 disks.  Three are shown in the following composite figure.  Note that the [[Apps/HayashiNaritaMiyama82#HNM82_Derivation|HNM82 flattening parameter]], <math>~\gamma</math>, has the following definition:
<div align="center">
<math>\gamma = [1 + v_\varphi^2/(2c_s^2)] \, .</math>
</div>
In practice, for real galaxy disks, this flattening parameter must be much much larger than 5 (the largest value I've shown in the composite figure) because <math>~V_\varphi</math> is on the order of 100 km/s while the sound speed is less than or on the order of one km/s; hence even the simplistic HNM82 models would be very thin indeed!

<table border="1" align="center" cellpadding="5" width="420px">
<tr>
  <th align="center">3D Renderings of Isothermal Disks from Hayashi, Narita &amp; Miyama (1982)</th>
</tr>
<tr><td align="center">
[[File:HNM82ThreeDisksAB.png|400px|Three HNM82 disks]]
</td></tr>
<tr>
  <td align="left">(Top) Edge-on view of three isothermal disks; (Bottom) Cut-away view of highest-density region of the same three disks.  Values of flattening parameter, <math>~\gamma</math>, shown in bottom-right corner of each frame.  
</table>


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<font color="red">Additional remarks by J. E. Tohline</font><p></p>
Date:  11 December 2015